Small Gradient Norm Regret for Online Convex Optimization

2026-01-20

AI Summary

This work addresses the limitation of existing regret measures in online convex optimization—such as $L^*$-regret—which become overly coarse when the curvature of loss functions approaches zero. To this end, the paper introduces a novel problem-dependent regret measure, termed $G^*$-regret, defined as the sum of squared norms of cumulative gradients evaluated at the post-hoc optimal decision point, tailored for smooth loss functions. This new measure strictly refines $L^*$-regret, yielding tighter bounds in low-curvature regimes, and naturally extends to dynamic regret and weighted feedback settings. Through rigorous upper and lower bound analyses complemented by empirical validation, the study demonstrates that under interpolation conditions, $G^*$-regret more accurately characterizes algorithmic performance, thereby significantly advancing the understanding of convergence behavior in both online and stochastic optimization.

Technology Category

• Machine Learning: Online Learning & Bandits
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper introduces a new problem-dependent regret measure for online convex optimization with smooth losses. The notion, which we call the $G^\star$ regret, depends on the cumulative squared gradient norm evaluated at the decision in hindsight. We show that the $G^\star$ regret strictly refines the existing $L^\star$ (small loss) regret, and that it can be arbitrarily sharper when the losses have vanishing curvature around the hindsight decision. We establish upper and lower bounds on the $G^\star$ regret and extend our results to dynamic regret and bandit settings. As a byproduct, we refine the existing convergence analysis of stochastic optimization algorithms in the interpolation regime. Some experiments validate our theoretical findings.

Problem

• online convex optimization
• regret minimization
• smooth losses
• gradient norm
• interpolation regime

Innovation

• G^star regret
• online convex optimization
• small gradient norm
• dynamic regret
• interpolation regime